# Saturated set with respect to a function

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## Contents

## Definition

Let [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] are sets and let [ilmath]f:X\rightarrow Y[/ilmath] be any function between them. A subset of [ilmath]X[/ilmath], [ilmath]U\in\mathcal{P}(X)[/ilmath], is said to be *saturated with respect to [ilmath]f[/ilmath]* if^{[1]}:

- [ilmath]\exists V\in\mathcal{P}(Y)[U=f^{-1}(V)][/ilmath], in words:
- There exists a subset of [ilmath]Y[/ilmath], [ilmath]V[/ilmath], such that [ilmath]V[/ilmath] is exactly the pre-image of [ilmath]U[/ilmath] under [ilmath]f[/ilmath]

## See next

## See also

- Fibre - this (saturated set) is a generalisation of a fibre.